compute_dxi_tilde

t3toolbox.backend.probing.compute_dxi_tilde(sigma_tildes, tau_tildes, down_tt_cores, mus, nus)

def compute_dxi_tilde(
        sigma_tildes:           typ.Union[typ.Sequence[NDArray], NDArray],  # len=d, elm_shape=(...,rR(i+1))
        tau_tildes:             typ.Union[typ.Sequence[NDArray], NDArray],  # len=d, elm_shape=(...,rLi)
        down_tt_cores:         typ.Union[typ.Sequence[NDArray], NDArray], # len=d, elm_shape=(rLi,nOi,rR(i+1))
        mus:                    typ.Union[typ.Sequence[NDArray], NDArray],  # len=d, elm_shape=(...,rLi)
        nus:                    typ.Union[typ.Sequence[NDArray], NDArray],  # len=d, elm_shape=(...,rR(i+1))
) -> typ.Union[typ.Sequence[NDArray], NDArray]: # dxi_tildes. len=d, elm_shape=(...,nOi)

Adjoint-var-downward edge variables dxi_tilde. Used for computing the transpose of the map from a tangent vector to its probes.

See Section 6.2.3, particularly Algorithm 8, in:

Alger, N., Christierson, B., Chen, P., & Ghattas, O. (2026). “Tucker Tensor Train Taylor Series.” arXiv preprint arXiv:2603.21141. https://arxiv.org/abs/2603.21141

Parameters:

• sigma_tildes (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray])

• tau_tildes (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray])

• down_tt_cores (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray])

• mus (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray])

• nus (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray])

Return type:

t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray]